Substrates formed of suitable solid-state materials may be used as platforms to support various structures, such as layered or graded panels, and multilevel, thin film microstructures deposited on the substrates. Integrated electronic circuits, integrated optical devices and opto-electronic circuits, micro-electro-mechanical systems (MEMS) deposited on wafers, three-dimensional electronic circuits, system-on-chip structures, lithographic reticles, and flat panel display systems (e.g., LCD and plasma displays) are examples of such structures integrated on various types of plate substrates. Substrates may be made of semiconductor materials (e.g., silicon wafers), silicon on insulator wafers (SOIs), amorphous or glass materials, polymeric or organic materials, and others. Different thin material layers or different structures may be formed on the same substrate in these structures and are in contact with one another to form various interfaces with adjacent structures and with the substrate. Some devices may use complex multilayer or continuously graded geometries. In addition, some devices may form various three dimensional structures.
The above and other structures on substrates may be made from a multiplicity of fabrication and processing steps and thus may experience stresses caused by these steps, such as deposition or thermal stresses. Examples of known phenomena and processes that build up stresses in thin films include but are not limited to, lattice mismatch, chemical reactions, doping by, e.g., diffusion or implantation, rapid deposition by evaporation or sputtering, and material removal (e.g. CMP or etch). As another example, a metallization process in fabricating integrated circuits may produce multiple layers on a semiconductor substrate (e.g., silicon), often at elevated temperatures. The multiple layers may include a mixture of metal and dielectric films which usually exhibit different mechanical, physical and thermal properties from those of the underlying substrate or wafer. Hence, such multiple layers can lead to high stresses in the film layers in the interconnection structures. These stresses can cause undesired stress-induced voiding in the metal interconnects and are directly related to electromigration. In addition, the stresses may cause cracking of some films and even delamination between various film layers, between interconnects and the encapsulating dielectrics, and between the films and the substrate. It is known that metal voiding, electromigration, cracking and delamination are among the leading causes for loss of subsequent process yield and failures in integrated circuits. Therefore, these and other stresses may adversely affect the structural integrity and operations of the structures or devices, and the lifetimes of such structures or devices. Hence, the identification of the origins of the stress build-up, the accurate measurement and analysis of stresses, and the acquisition of information on the spatial distribution of such stresses are important in designing and processing the structures or devices and to improving the reliability and manufacturing yield of various layered structures.
Stresses in layered thin-film structures deposited on plate substrates may be calculated from the substrate curvature or “bow” based on a correlation between changes in the curvature and stress changes at the same location. Early attempts to provide such correlation are well known. Various formulations have been developed for measurements of stresses in thin films and most of these formulations are essentially based on extensions of Stoney's approximate plate analysis published in Proceedings of the Royal Society, London, Series A, vol. 82, pp. 172(1909). Stoney studied a thin film deposited on a relatively thick substrate, and derived the Stoney formula to give the relation between the system curvature κ and the film stress σ(f),
                                          σ                          (              f              )                                =                                                    E                s                            ⁢                              h                s                2                            ⁢              κ                                      6              ⁢                                                h                  f                                ⁡                                  (                                      1                    -                                          v                      s                                                        )                                                                    ,                            (        1.1        )            where the subscripts “f” and “s” denote the thin film and substrate, respectively, and E, v and h are the Young's modulus, Poisson's ratio and thickness. Equation (1.1) has been extensively used to infer film stress changes from experimental measurement of system curvature changes.
Stoney's formula was based on the following assumptions, some of which have been relaxed: (i) Both the film thickness hf and the substrate thickness hs are uniform and hf<<hs<<R, where R represents the characteristic length in the lateral direction (e.g., system radius R shown in FIG. 1). This assumption was recently relaxed for the thin film and substrate of different radii, arbitrarily non-uniform film thickness and substrate thickness. These analytical results have been verified by the X-ray microdiffraction experiments; (ii) The strains and rotations of the plate system are infinitesimal. This assumption has been relaxed by various “large” deformation analyses, some of which have been validated by experiments; (iii) Both the film and substrate are homogeneous, isotropic, and linearly elastic. To our best knowledge this assumption has not been relaxed yet. (iv) The film stress states are equi-biaxial (σxx=σyy,σxy=0) while the out-of-plane direct stress and all shear stresses vanish (σzz=σxz=σyz=0). This assumption has been relaxed for non-equi-biaxial but constant stress states, and recently to non-equi-biaxial and spatially non-uniform stress states; (v) The system's curvature components are equi-biaxial (κxx=κyy) while the twist curvature vanishes κxy=0. This assumption has been relaxed for non-equi-biaxial but constant curvature components and non-vanishing (but constant) twist components, and recently for non-equi-biaxial and non-constant curvature components; (vi) All surviving stress and curvature components are spatially constant over the plate system's surface, a situation that is often violated in practice. Recently this assumption was relaxed for the thin film/substrate system subjected to non-uniform, axisymmetric misfit strain (in thin film) and temperature change (in both thin film and substrate), respectively, while the thin film/substrate system subject to arbitrarily non-uniform (e.g., non-axisymmetric) misfit strain and temperature was also studied. Their most important result is that the film stresses depend non-locally on the system curvatures, i.e., they depend on curvatures of the entire system.
Despite the explicitly stated assumptions of spatial stress and curvature uniformity, the Stoney formula is often, arbitrarily, applied to cases of practical interest where these assumptions are violated. This is typically done by applying Stoney's formula pointwise and thus extracting a local value of stress from a local measurement of the curvature of the system. This approach of inferring film stress clearly violates the uniformity assumptions of the analysis and, as such, its accuracy as an approximation deteriorates as the levels of curvature non-uniformity become more severe. For example, X-ray diffraction experiments showed that the Stoney formula may underestimate the film stress by 50%.
Many thin film/substrate systems involve multiple layers of thin films. The Stoney formula was extended for multi-layer thin films subjected to non-uniform misfit strains and temperature, respectively. But they still assumed constant thickness of each film. But in applications the multi-layer thin films often have non-constant thickness, such as films forming an “island” on the substrate. Current methodologies used for the inference of thin film stress through curvature measurement are strictly restricted to stress and curvature states that are assumed to remain uniform over the entire film/substrate system. These methodologies have recently been extended to a single thin film of non-uniform thickness deposited on a substrate subjected to the non-uniform misfit strain in the thin film.